ABCD is a quadrilateral such that BC = BA and `CD gt AD`. Which one of the following is correct ?

Options

A. `angle BAX = angle BCD`
B. `angle BAD lt angle BCD`
C. `angle BAD gt angle BCD`
D. `2angle BAD = angle BCD`

Answer

To solve the problem, we need to analyze the properties of the quadrilateral ABCD based on the given conditions: BC = BA and CD > AD.

### Step-by-Step Solution:

1. **Draw the Quadrilateral**:
- First, sketch the quadrilateral ABCD. Label the points A, B, C, and D appropriately.

2. **Identify the Given Conditions**:
- We know that BC = BA, which means triangle ABC is isosceles with AB = BC.
- We also know that CD > AD.

3. **Analyze Triangle ABC**:
- Since AB = BC, the angles opposite these sides must be equal. Therefore, angle BAC = angle BCA.

4. **Analyze Triangle ADC**:
- In triangle ADC, since CD > AD, the angle opposite to CD (which is angle DAC) must be greater than the angle opposite to AD (which is angle DCA). Thus, angle DAC > angle DCA.

5. **Compare Angles**:
- We have established that angle BAC = angle BCA from triangle ABC.
- We also know that angle DAC > angle DCA from triangle ADC.
- Since angle DAC is greater than angle DCA, and angle BAC is equal to angle BCA, we can conclude that angle BAD (which is equal to angle BAC) is greater than angle BCD (which is equal to angle DCA).

6. **Conclusion**:
- Therefore, we conclude that angle BAD > angle BCD.

### Final Answer:
The correct statement is that angle BAD is greater than angle BCD.

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